Suppose you're stuck in the middle of a frozen lake, with a surface so slippery that you can't get any traction. You take off a shoe and throw it toward the southern shore. You find yourself coasting toward the northern shore and soon escape from the lake. Why did this scheme work?

In order to understand the situation with the frozen lake, let's delve into Newton's third law of motion. This law tells us that for every action, there is an equal and opposite reaction. That means if you exert a force on an object, that object exerts an equal force back on you.
For example, when you throw your shoe toward the southern shore, you are applying a force to the shoe in the southern direction. According to Newton's third law, the shoe applies an equal and opposite force back on you. This force is what pushes you toward the northern shore.
It's crucial to remember that these forces are always equal in magnitude and opposite in direction. Therefore, even though the action (throwing the shoe) seems like the primary movement, the reaction (being pushed backward) is just as important and immediately follows.

Now, let's consider the frictionless surface of the frozen lake. Friction is the force that resists the motion of objects sliding against each other. On the ice, this friction is negligible or even nonexistent. When you try to walk or move on a frictionless surface, there's nothing for your feet to push against and you can't generate any traction.
This lack of traction is why you can't just walk to the shore. When you threw your shoe, however, the frictionless surface actually helped you. Without friction to slow you down, the small force you exerted by throwing the shoe was enough to move you towards the shore.
Consequently, the slippery surface becomes an advantage, simplifying the scenario to just the forces from throwing the shoe and nothing else counteracting your movement.

Lastly, let's explore the general physics of motion involved in this process. Motion is governed by Newton's laws, which describe how forces are applied and how objects move. When you threw your shoe, you turned potential energy (stored in your muscles) into kinetic energy in the shoe and yourself.
Kinetic energy is the energy an object has because of its motion. Because the ice is frictionless, both you and the shoe continue to move without any substantial hindrance.
Understanding this conversion of energy and the continuity of motion helps us see why you moved towards the shore. In summary, the action of throwing the shoe led to the reaction of moving yourself, and the frictionless surface ensured nothing interrupted your motion toward the shoreline.